Finite Group Databasebeta

← #10690848

Subgroups — #10690848

Order 768.

588 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2857C211yesmaximal, characteristic
2C2×#4925C211yesmaximal, characteristic
3C2×#4547C212yesmaximal, complemented
4#71530C211yesmaximal, characteristic
5#67634C212yesmaximal, complemented
6#9132—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×C3×C22⋊Q8—21no—
9C22×C3×C42C2211yescharacteristic
10C3×#391—21no—
11C3×#389—21no—
12C3×#389C412yes—
13C3×#384C2211yescharacteristic
14C3×#380—21no—
15C3×#380C412yes—
16C2×#830—21no—
17C2×#830—21no—
18#4925—24no—
19C2×C4⋊(C3⋊C8)C2212yes—
20C2×C4⋊(C3⋊C8)C2212yes—
21C2×C4×C3⋊C8—21no—
22#4547—24no—
23#4524—21no—
24C2×#431—31no—
25#2857D311yescharacteristic
26#1797—31no—
27C2×#332—32no—
28#1296—32no—
29C22×C3×Q8—21no—
30C23×C3×C4C2×C411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C3×C4⋊Q8—22no—
34C3×C4⋊Q8—22no—
35C3×C22⋊Q8—44no—
36C2×C3×C4⋊C4—22no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4C2×C412yes—
41C2×C3×C4⋊C4C2×C412yes—
42C2×C3×C22⋊C4—22no—
43C2×C3×C42—22no—
44C2×C3×C42—22no—
45C2×C3×C42C2×C411yescharacteristic
46C2×C3×C42D411yescharacteristic
47C2×C3×C42D411yescharacteristic
48C2×C3×C42C2311yescharacteristic
49C22×C3⋊C8—22no—
50C22×C3⋊C8—22no—
51C3×#95—22no—
52C3×#95—22no—
53#830—24no—
54#830—42no—
55C4⋊(C3⋊C8)—22no—
56C4⋊(C3⋊C8)—22no—
57C4⋊(C3⋊C8)—22no—
58C4⋊(C3⋊C8)D414yes—
59C4×C3⋊C8—24no—
60C2×C4⋊Q8C2×D311yescharacteristic
61C2×C22⋊Q8—21no—
62C22×C42C2×D311yescharacteristic
63#431—64no—
64C2×C4⋊C8—32no—
65C2×C4⋊C8—32no—
66C2×C22⋊C8—61no—
67C2×C22⋊C8—61no—
68C2×C4×C8—61no—
69#391—21no—
70#389—21no—
71#389Dic312yes—
72#384C2×D311yescharacteristic
73#380—21no—
74#380Dic312yes—
75#336—61no—
76#332—64no—
77C24×C3C22⋊C411yescharacteristic
78C2×C3×Q8—41no—
79C2×C3×Q8—41no—
80C2×C3×Q8—42no—
81C2×C3×Q8—42no—
82C22×C3×C4—22no—
83C22×C3×C4—22no—
84C22×C3×C4—22no—
85C22×C3×C4—22no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—22no—
89C22×C3×C4—22no—
90C22×C3×C4—22no—
91C22×C3×C4—22no—
92C22×C3×C4—41no—
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22⋊C411yescharacteristic
95C22×C3×C4C22⋊C411yescharacteristic
96C22×C3×C4SD1612yes—
97C22×C3×C4Q1612yes—
98C22×C3×C4C22×C411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C3×C4⋊C4—24no—
102C3×C4⋊C4—42no—
103C3×C4⋊C4—44no—
104C3×C4⋊C4—42no—
105C3×C4⋊C4—42no—
106C3×C4⋊C4—42no—
107C3×C22⋊C4—44no—
108C3×C42—22no—
109C3×C42—21no—
110C3×C42—22no—
111C3×C42—21no—
112C3×C42C2×D412yes—
113C3×C42#4112yes—
114C2×C3⋊C8—24no—
115C2×C3⋊C8—42no—
116C2×C3⋊C8—42no—
117C2×C3⋊C8—42no—
118C2×C3⋊C8—42no—
119C2×C3⋊C8—42no—
120C22×Q8—21no—
121C23×C4C2×Dic311yescharacteristic
122C23×C4#6711yescharacteristic
123C23×C4#6711yescharacteristic
124C22×C8—62no—
125C22×C8—62no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C22⋊Q8—44no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4C2×Dic312yes—
134C2×C4⋊C4C2×Dic312yes—
135C2×C22⋊C4—22no—
136C2×C42—22no—
137C2×C42—22no—
138C2×C42C2×Dic311yescharacteristic
139C2×C42#6711yescharacteristic
140C2×C42#6711yescharacteristic
141C2×C42C22×D311yescharacteristic
142C4⋊C8—34no—
143C4⋊C8—62no—
144C4⋊C8—62no—
145C4⋊C8—62no—
146C22⋊C8—64no—
147C22⋊C8—122no—
148C4×C8—64no—
149#95—22no—
150#95—22no—
151C23×C3—21no—
152C23×C3—21no—
153C23×C3—21no—
154C23×C3—21no—
155C23×C3—21no—
156C23×C3—21no—
157C23×C3#10312yes—
158C23×C3C2×C22⋊C411yescharacteristic
159C3×Q8—82no—
160C3×Q8—82no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—24no—
168C2×C3×C4—24no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—44no—
173C2×C3×C4—42no—
174C2×C3×C4—42no—
175C2×C3×C4—42no—
176C2×C3×C4—42no—
177C2×C3×C4—42no—
178C2×C3×C4—42no—
179C2×C3×C4—42no—
180C2×C3×C4—42no—
181C2×C3×C4#10112yes—
182C2×C3×C4#10312yes—
183C2×C3×C4C4×D412yes—
184C2×C3×C4#12011yescharacteristic
185C2×C3×C4#12112yes—
186C2×C3×C4#12311yescharacteristic
187C2×C3×C4C2×SD1611yescharacteristic
188C2×C3×C4C2×Q1611yescharacteristic
189C2×C3×C4#13512yes—
190C2×C3×C4#13512yes—
191C2×C3×C4AGL(1,ℤ/8)12yes—
192C3⋊C8—44no—
193C3⋊C8—82no—
194C24#21711yescharacteristic
195C2×Q8—42no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—42no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—41no—
210C22×C4#21512yes—
211C22×C4#21612yes—
212C22×C4#21711yescharacteristic
213C22×C4#21711yescharacteristic
214C22×C4#21711yescharacteristic
215C22×C4C22×Dic311yescharacteristic
216C22×C4C2×#6711yescharacteristic
217C22×C4C2×#6711yescharacteristic
218C2×C8—64no—
219C2×C8—122no—
220C2×C8—122no—
221C2×C8—122no—
222C2×C8—122no—
223C2×C8—122no—
224C4⋊C4—24no—
225C4⋊C4—42no—
226C4⋊C4—44no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C22⋊C4—44no—
231C42—22no—
232C42—21no—
233C42—21no—
234C42—22no—
235C42D3×D412yes—
236C42#23712yes—
237C22×C3—22no—
238C22×C3—22no—
239C22×C3—22no—
240C22×C3—41no—
241C22×C3—41no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3#38611yescharacteristic
245C22×C3C2×#10111yescharacteristic
246C22×C3#41311yescharacteristic
247C22×C3C2×#10311yescharacteristic
248C22×C3#41611yescharacteristic
249C22×C3#41611yescharacteristic
250C22×C3#41811yescharacteristic
251C3×C4—22no—
252C3×C4—22no—
253C3×C4—22no—
254C3×C4—41no—
255C3×C4—82no—
256C3×C4—82no—
257C3×C4#44912yes—
258C3×C4#46512yes—
259C3×C4#46712yes—
260C3×C4#48212yes—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23#83512yes—
268C23C2×#21711yescharacteristic
269Q8—82no—
270Q8—82no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4#83512yes—
292C2×C4#83612yes—
293C2×C4#93212yes—
294C2×C4D4×Dic312yes—
295C2×C4#93511yescharacteristic
296C2×C4#93711yescharacteristic
297C2×C4#93912yes—
298C2×C4C2×#21511yescharacteristic
299C2×C4C2×#21611yescharacteristic
300C2×C4#95012yes—
301C2×C4#95012yes—
302C8—124no—
303C8—242no—
304C2×C3—42no—
305C2×C3#154211yescharacteristic
306C2×C3#154411yescharacteristic
307C2×C3#154911yescharacteristic
308C2×C3#155311yescharacteristic
309C2×C3#187511yescharacteristic
310C2×C3#189211yescharacteristic
311C2×C3#190111yescharacteristic
312C22—22no—
313C22—22no—
314C22—22no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22#521511yescharacteristic
320C22#522011yescharacteristic
321C22C2×#83511yescharacteristic
322C22C2×#83611yescharacteristic
323C22#523311yescharacteristic
324C22#523311yescharacteristic
325C22#523711yescharacteristic
326C4—22no—
327C4—22no—
328C4—22no—
329C4—41no—
330C4—82no—
331C4—82no—
332C4#503512yes—
333C4#503712yes—
334C4#504812yes—
335C4#505112yes—
336C3#913211yescharacteristic, complemented, Sylow
337C2—42no—
338C2#7167811yescharacteristic
339C2#7169611yescharacteristic
340C2#7169811yescharacteristic
341C2#7177511yescharacteristic
342C2#7180811yescharacteristic
343C2#7182811yescharacteristic
344C2#7182911yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2857C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2857D3
hypocenterC3#9132
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#4547C2semidirect
5. #67634C2semidirect
336. C3#9132semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#28572
C2×#49252
C2×#45472
#715302
#676342
#91323